Problem: Let $y=\csc(x)$. What is the value of $\dfrac{dy}{dx}$ at $x=\dfrac{\pi}{3}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac32$ (Choice B) B $-\dfrac{3}{8}$ (Choice C) C $-\dfrac23$ (Choice D) D $\dfrac{8}{3}$
Let's first find $\dfrac{dy}{dx}$. Then, we can evaluate it at $x=\dfrac{\pi}{3}$. Recall that the derivative of $\csc(x)$ is $-\dfrac{\cos(x)}{\sin^2(x)}$, or $-\csc(x)\cot(x)$. Put another way, $\dfrac{d}{dx}[\csc(x)]=-\dfrac{\cos(x)}{\sin^2(x)}=-\csc(x)\cot(x)$. [Is there a way to know this without memorizing?] Now let's plug in $x={\dfrac{\pi}{3}}$ : $\begin{aligned} &\phantom{=}-\dfrac{\cos\left({\dfrac{\pi}{3}}\right)}{\sin^2\left({\dfrac{\pi}{3}}\right)} \\\\ &=-\dfrac{\dfrac12}{\left(\dfrac{\sqrt{3}}{2}\right)^2} \\\\ &=-{\dfrac12}\cdot{\left(\dfrac{4}{3}\right)} \\\\ &=-\dfrac23 \end{aligned}$ In conclusion, the value of $\dfrac{dy}{dx}$ at $x=\dfrac{\pi}{3}$ is $-\dfrac23$.